randist.texi

开放gsl矩阵运算

This function returns a random variate from the Landau distribution.  The
probability distribution for Landau random variates is defined
analytically by the complex integral,

@tex
\beforedisplay
$$
p(x) = 
{1 \over {2 \pi i}} \int_{c-i\infty}^{c+i\infty} ds\, \exp(s \log(s) + x s) 
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) = (1/(2 \pi i)) \int_@{c-i\infty@}^@{c+i\infty@} ds exp(s log(s) + x s) 
@end example
@end ifinfo
For numerical purposes it is more convenient to use the following
equivalent form of the integral,
@tex
\beforedisplay
$$
p(x) = (1/\pi) \int_0^\infty dt \exp(-t \log(t) - x t) \sin(\pi t).
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) = (1/\pi) \int_0^\infty dt \exp(-t \log(t) - x t) \sin(\pi t).
@end example
@end ifinfo
@end deftypefn

@deftypefun double gsl_ran_landau_pdf (double @var{x})
This function computes the probability density @math{p(x)} at @var{x}
for the Landau distribution using an approximation to the formula given
above.
@end deftypefun

@sp 1
@tex
\centerline{\input rand-landau.tex}
@end tex

@page
@node The Levy alpha-Stable Distributions
@section The Levy alpha-Stable Distributions
@deftypefn Random double gsl_ran_levy (const gsl_rng * @var{r}, double @var{c}, double @var{alpha})
@cindex Levy distribution, random variates
This function returns a random variate from the Levy symmetric stable
distribution with scale @var{c} and exponent @var{alpha}.  The symmetric
stable probability distribution is defined by a fourier transform,

@tex
\beforedisplay
$$
p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^\alpha)
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) = @{1 \over 2 \pi@} \int_@{-\infty@}^@{+\infty@} dt \exp(-it x - |c t|^alpha)
@end example
@end ifinfo
@noindent
There is no explicit solution for the form of @math{p(x)} and the
library does not define a corresponding @code{pdf} function.  For
@math{\alpha = 1} the distribution reduces to the Cauchy distribution.  For
@math{\alpha = 2} it is a Gaussian distribution with @c{$\sigma = \sqrt{2} c$} 
@math{\sigma = \sqrt@{2@} c}.  For @math{\alpha < 1} the tails of the
distribution become extremely wide.

The algorithm only works for @c{$0 < \alpha \le 2$}
@math{0 < alpha <= 2}.
@end deftypefn

@sp 1
@tex
\centerline{\input rand-levy.tex}
@end tex

@page
@node The Levy skew alpha-Stable Distribution
@section The Levy skew alpha-Stable Distribution

@deftypefn Random double gsl_ran_levy_skew (const gsl_rng * @var{r}, double @var{c}, double @var{alpha}, double @var{beta})
@cindex Levy distribution, random variates
This function returns a random variate from the Levy skew stable
distribution with scale @var{c}, exponent @var{alpha} and skewness
parameter @var{beta}.  The skewness parameter must lie in the range
@math{[-1,1]}.  The Levy skew stable probability distribution is defined
by a fourier transform,

@tex
\beforedisplay
$$
p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^\alpha (1-i \beta sign(t) \tan(\pi\alpha/2)))
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) = @{1 \over 2 \pi@} \int_@{-\infty@}^@{+\infty@} dt \exp(-it x - |c t|^alpha (1-i beta sign(t) tan(pi alpha/2)))
@end example
@end ifinfo
@noindent
When @math{\alpha = 1} the term @math{\tan(\pi \alpha/2)} is replaced by
@math{-(2/\pi)\log|t|}.  There is no explicit solution for the form of
@math{p(x)} and the library does not define a corresponding @code{pdf}
function.  For @math{\alpha = 2} the distribution reduces to a Gaussian
distribution with @c{$\sigma = \sqrt{2} c$} 
@math{\sigma = \sqrt@{2@} c} and the skewness parameter has no effect.  
For @math{\alpha < 1} the tails of the distribution become extremely
wide.  The symmetric distribution corresponds to @math{\beta =
0}.

The algorithm only works for @c{$0 < \alpha \le 2$}
@math{0 < alpha <= 2}.
@end deftypefn

The Levy alpha-stable distributions have the property that if @math{N}
alpha-stable variates are drawn from the distribution @math{p(c, \alpha,
\beta)} then the sum @math{Y = X_1 + X_2 + \dots + X_N} will also be
distributed as an alpha-stable variate,
@c{$p(N^{1/\alpha} c, \alpha, \beta)$}  
@math{p(N^(1/\alpha) c, \alpha, \beta)}.

@comment PDF not available because there is no analytic expression for it
@comment
@comment @deftypefun double gsl_ran_levy_pdf (double @var{x}, double @var{mu})
@comment This function computes the probability density @math{p(x)} at @var{x}
@comment for a symmetric Levy distribution with scale parameter @var{mu} and
@comment exponent @var{a}, using the formula given above.
@comment @end deftypefun

@sp 1
@tex
\centerline{\input rand-levyskew.tex}
@end tex

@page
@node The Gamma Distribution
@section The Gamma Distribution
@deftypefn Random double gsl_ran_gamma (const gsl_rng * @var{r}, double @var{a}, double @var{b})
@cindex Gamma distribution random variates
This function returns a random variate from the gamma
distribution.  The distribution function is,

@tex
\beforedisplay
$$
p(x) dx = {1 \over \Gamma(a) b^a} x^{a-1} e^{-x/b} dx
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) dx = @{1 \over \Gamma(a) b^a@} x^@{a-1@} e^@{-x/b@} dx
@end example
@end ifinfo
@noindent
for @math{x > 0}.
@comment If @xmath{X} and @xmath{Y} are independent gamma-distributed random
@comment variables of order @xmath{a} and @xmath{b}, then @xmath{X+Y} has a gamma
@comment distribution of order @xmath{a+b}.
@end deftypefn

@deftypefun double gsl_ran_gamma_pdf (double @var{x}, double @var{a}, double @var{b})
This function computes the probability density @math{p(x)} at @var{x}
for a gamma distribution with parameters @var{a} and @var{b}, using the
formula given above.
@end deftypefun

@sp 1
@tex
\centerline{\input rand-gamma.tex}
@end tex

@page
@node The Flat (Uniform) Distribution
@section The Flat (Uniform) Distribution
@deftypefn Random double gsl_ran_flat (const gsl_rng * @var{r}, double @var{a}, double @var{b})
@cindex flat distribution random variates
This function returns a random variate from the flat (uniform)
distribution from @var{a} to @var{b}. The distribution is,

@tex
\beforedisplay
$$
p(x) dx = {1 \over (b-a)} dx
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) dx = @{1 \over (b-a)@} dx
@end example
@end ifinfo
@noindent
if @c{$a \le x < b$}
@math{a <= x < b} and 0 otherwise.
@end deftypefn

@deftypefun double gsl_ran_flat_pdf (double @var{x}, double @var{a}, double @var{b})
This function computes the probability density @math{p(x)} at @var{x}
for a uniform distribution from @var{a} to @var{b}, using the formula
given above.
@end deftypefun

@sp 1
@tex
\centerline{\input rand-flat.tex}
@end tex

@page
@node The Lognormal Distribution
@section The Lognormal Distribution
@deftypefn Random double gsl_ran_lognormal (const gsl_rng * @var{r}, double @var{zeta}, double @var{sigma})
@cindex Lognormal random variates
This function returns a random variate from the lognormal
distribution.  The distribution function is,

@tex
\beforedisplay
$$
p(x) dx = {1 \over x \sqrt{2 \pi \sigma^2}} \exp(-(\ln(x) - \zeta)^2/2 \sigma^2) dx
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) dx = @{1 \over x \sqrt@{2 \pi \sigma^2@} @} \exp(-(\ln(x) - \zeta)^2/2 \sigma^2) dx
@end example
@end ifinfo
@noindent
for @math{x > 0}.
@end deftypefn

@deftypefun double gsl_ran_lognormal_pdf (double @var{x}, double @var{zeta}, double @var{sigma})
This function computes the probability density @math{p(x)} at @var{x}
for a lognormal distribution with parameters @var{zeta} and @var{sigma},
using the formula given above.
@end deftypefun

@sp 1
@tex
\centerline{\input rand-lognormal.tex}
@end tex

@page
@node The Chi-squared Distribution
@section The Chi-squared Distribution
The chi-squared distribution arises in statistics If @math{Y_i} are
@math{n} independent gaussian random variates with unit variance then the
sum-of-squares,

@tex
\beforedisplay
$$
X_i = \sum_i Y_i^2
$$
\afterdisplay
@end tex
@ifinfo
@example
X_i = \sum_i Y_i^2
@end example
@end ifinfo
@noindent
has a chi-squared distribution with @math{n} degrees of freedom.

@deftypefn Random double gsl_ran_chisq (const gsl_rng * @var{r}, double @var{nu})
@cindex Chi-squared random variates
This function returns a random variate from the chi-squared distribution
with @var{nu} degrees of freedom. The distribution function is,

@tex
\beforedisplay
$$
p(x) dx = {1 \over \Gamma(\nu/2) } (x/2)^{\nu/2 - 1} \exp(-x/2) dx
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) dx = @{1 \over \Gamma(\nu/2) @} (x/2)^@{\nu/2 - 1@} \exp(-x/2) dx
@end example
@end ifinfo

@noindent
for @c{$x \ge 0$}
@math{x >= 0}. 
@end deftypefn

@deftypefun double gsl_ran_chisq_pdf (double @var{x}, double @var{nu})
This function computes the probability density @math{p(x)} at @var{x}
for a chi-squared distribution with @var{nu} degrees of freedom, using
the formula given above.
@end deftypefun

@sp 1
@tex
\centerline{\input rand-chisq.tex}
@end tex

@page
@node The F-distribution
@section The F-distribution
The F-distribution arises in statistics.  If @math{Y_1} and @math{Y_2}
are chi-squared deviates with @math{\nu_1} and @math{\nu_2} degrees of
freedom then the ratio,

@tex
\beforedisplay
$$
X = { (Y_1 / \nu_1) \over (Y_2 / \nu_2) }
$$
\afterdisplay
@end tex
@ifinfo
@example
X = @{ (Y_1 / \nu_1) \over (Y_2 / \nu_2) @}
@end example
@end ifinfo
@noindent
has an F-distribution @math{F(x;\nu_1,\nu_2)}.

@deftypefn Random double gsl_ran_fdist (const gsl_rng * @var{r}, double @var{nu1}, double @var{nu2})
@cindex F-distribution random variates
This function returns a random variate from the F-distribution with degrees of freedom @var{nu1} and @var{nu2}. The distribution function is,

@tex
\beforedisplay
$$
p(x) dx = 
   { \Gamma((\nu_1 + \nu_2)/2)
        \over \Gamma(\nu_1/2) \Gamma(\nu_2/2) } 
   \nu_1^{\nu_1/2} \nu_2^{\nu_2/2} 
   x^{\nu_1/2 - 1} (\nu_2 + \nu_1 x)^{-\nu_1/2 -\nu_2/2}
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) dx = 
   @{ \Gamma((\nu_1 + \nu_2)/2)
        \over \Gamma(\nu_1/2) \Gamma(\nu_2/2) @} 
   \nu_1^@{\nu_1/2@} \nu_2^@{\nu_2/2@} 
   x^@{\nu_1/2 - 1@} (\nu_2 + \nu_1 x)^@{-\nu_1/2 -\nu_2/2@}
@end example
@end ifinfo

@noindent
for @c{$x \ge 0$}
@math{x >= 0}. 
@end deftypefn

@deftypefun double gsl_ran_fdist_pdf (double @var{x}, double @var{nu1}, double @var{nu2})
This function computes the probability density @math{p(x)} at @var{x}
for an F-distribution with @var{nu1} and @var{nu2} degrees of freedom,
using the formula given above.
@end deftypefun

@sp 1
@tex
\centerline{\input rand-fdist.tex}
@end tex

@page
@node The t-distribution
@section The t-distribution
The t-distribution arises in statistics.  If @math{Y_1} has a normal
distribution and @math{Y_2} has a chi-squared distribution with
@math{\nu} degrees of freedom then the ratio,

@tex
\beforedisplay
$$
X = { Y_1 \over \sqrt{Y_2 / \nu} }
$$
\afterdisplay
@end tex
@ifinfo
@example
X = @{ Y_1 \over \sqrt@{Y_2 / \nu@} @}
@end example
@end ifinfo

@noindent
has a t-distribution @math{t(x;\nu)} with @math{\nu} degrees of freedom.

@deftypefn Random double gsl_ran_tdist (const gsl_rng * @var{r}, double @var{nu})
@cindex t-distribution random variates
This function returns a random variate from the t-distribution.  The
distribution function is,

@tex
\beforedisplay
$$
p(x) dx = {\Gamma((\nu + 1)/2) \over \sqrt{\pi \nu} \Gamma(\nu/2)}
   (1 + x^2/\nu)^{-(\nu + 1)/2} dx
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) dx = @{\Gamma((\nu + 1)/2) \over \sqrt@{\pi \nu@} \Gamma(\nu/2)@}
   (1 + x^2/\nu)^@{-(\nu + 1)/2@} dx
@end example
@end ifinfo
@noindent
for @math{-\infty < x < +\infty}.
@end deftypefn

@deftypefun double gsl_ran_tdist_pdf (double @var{x}, double @var{nu})
This function computes the probability density @math{p(x)} at @var{x}
for a t-distribution with @var{nu} degrees of freedom, using the formula
given above.
@end deftypefun

@sp 1
@tex
\centerline{\input rand-tdist.tex}
@end tex

@page
@node The Beta Distribution
@section The Beta Distribution
@deftypefn Random double gsl_ran_beta (const gsl_rng * @var{r}, double @var{a}, double @var{b})
@cindex Beta distribution random variates
This function returns a random variate from the beta
distribution.  The distribution function is,

@tex
\beforedisplay
$$
p(x) dx = {\Gamma(a+b) \over \Gamma(a) \Gamma(b)} x^{a-1} (1-x)^{b-1} dx
$$
\afterdisplay
@end tex
@ifinfo
@example
p(x) dx = @{\Gamma(a+b) \over \Gamma(a) \Gamma(b)@} x^@{a-1@} (1-x)^@{b-1@} dx
@end example
@end ifinfo
@noindent